直接计算自然数的方幂和的一种方法

计算前n个自然数的方幂和的方法,在国内书刊上已有介绍(例如,上海教育出版社的《初等数学论丛》第7辑上就有《计算自然数的方幂和的一种方法》一文).这些文章所介绍的方法,大多采用“递推”方式给出的,即是,欲求前n个自然数的m次方幂和,得先算出前n个自然数的1次方幂和,2次方幂和……,m—1次方幂和,在此基础上,再用所述方法计算m次     

倒序法求前n个自然数的平方和

关于求解前n个自然数平方和问题,至少已经有10多种方法了.其中,最经典的是利用n3－(n－1) 3求解.因为这种解法不但解题快速,更重要的是可以推广求解自然数方幂和.     

等差数列的两个性质及其应用

等差数列 {an}中 ,任意两项 an、am 存在关系 :an =am + ( n - m) d,利用此式 ,有时解题非常简捷、迅速 ,这个性质我们都很熟悉 .由此 ,猜想 :等差数列中 ,前 n项和 Sn与前 m项和Sm 之间 ,Sn 与 an 之间 ,是否也存在一种关系呢 ?这种关系在解题时 ,是否能给我们带来方便 ?本文将探讨这个问题 .由等差数列的通项公式am =a1 + ( m - 1 ) d,得 a1 =am+ ( 1 - m) d,代入　 Sn =na1 + n( n - 1 ) d2 ,得　 Sn =n[am + ( 1 - m) d]+ n( n - 1 ) d2=nam + n( n + 1 - 2 m) d2 ( 1 )公式 ( 1 )反映了等差数列前 n项和与其任一项之间的关系 .由 ( 1…     

正项二阶等差数列的不等式

文[1]、[2]研究了正项等差数列方幂的不等式，本文研究由递增正项二阶等差数列若干项构成的不等式，为了简便起见，以下约定{an}是递增正项二阶等差数列，bn=a（n＋1）-an，{bn}的公差为d，其前n项和为Sn，m，k，n，p为正整数．     

自然数幂和公式的另一种计算机实现方法

自然数幂和问题,许多文献给出不同算法,借助M ATLAB系统,利用自然数幂和的矩阵算法,给出可计算所有小于任意指定自然数m的自然数幂和公式的计算机实现方法.     

求自然数幂和的差分公式法

给出了求自然数幂和的一种差分公式法.首先将自然数幂和表示为有限项多项式形式,且每一项为关于自然数n的递减连乘积,由此建立以各项系数为未知量的线性方程组,并通过应用矩阵初等变换以及差分算子的计算工具,从而得到求自然数幂和的又一计算公式.     

用线性方程组与Mathematica软件求解两类自然数幂和公式

利用三个线性方程组与M athem atica4.0软件,给出求解古典自然数幂和公式Sk(n)(k 0,n∈N+)与现代自然数幂和公式Tk(n)(k 0,n∈N+)的若干新的机械计算方法.     

等差与等比数列不等式的互变

含有等差或等比数列若干项的不等式 ,为行文方便不妨叫做等差或等比数列不等式 .本文研究这两种不等式的互变 .为了叙述简便 ,本文规定数列 {an}是公差为d的等差数列 ,其前n项的和为Sn,数列 {bn}是公比为 q的等比数列 ,其前n项的积为Tn,m ,n ,k是互不相等的正自然数 .通过下面等差与等比数列互换表中的an 与bn 等的互换 ,能够实现这两种不等式的互变 ,但互换两种运算时 ,应注意它们的基本要求 .　　引理 1　若mk =n2 ,则m +k >2n .证　m +k >2mk =2n2 =2n .引理 2　若m +k =2n ,则mk 相似文献   

正项二阶等差数列的不等式

文[1]、[2]研究了正项等差数列方幂的不等式,本文研究由递增正项二阶等差数列若干项构成的不等式.为了简便起见,以下约定{an}是递增正项二阶等差数列,bn=an 1-an,{bn}的公差为d,其前n项和为Sn,m,k,n,p为正整数.引理d>0,an 1=Sn a1.证设an=an2 bn c,a,b,c∈R,且a>0.∵bn=an 1-an     

利用等差中项巧解数列问题

请先看下面的问题: 设有二等差数列{αn}、{bn},其前n项和Sn与Sn’之比为了5n 3/2n 7,求α9/b9的值. 在对此题的分析和研究中,我们发现等差数列的等差中项有一个简单性质,它可以加以推广应用.请看: 设{αn}是一个等差数列,Sn是其前n项和,则有: S3=3α2,S5=5α3,S7=7α4,…… 一般地有: S2n-1=(2n-1)αn,其中n是自然数. 要证明此结论很简单.根据等差数列的前n项和公式及通项公式得:     

Difference Equations and Divisibility Properties of Sequences

There are many different ways of defining a sequence in terms of solutions to difference equations. In fact, if a sequence satisfies one recurrence then it satisfies an infinite number of recurrences. Arithmetic properties of an integral sequence are often studied by direct methods based on the combinatorial or algebraic definition of the numbers or using their generating function. The rational generating function is the main tool in obtaining various difference equations with coefficients and initial values exhibiting divisibility patterns that can imply particular arithmetic properties of the solutions. In this process, we face the challenging task of finding difference equations that are relevant to the divisibility properties by transforming the original rational generating function. As a matter of fact, it is not necessarily the simple difference equation that helps the most in proving the properties. We illustrate this process on several examples and a sequence involving a p -sected binomial sum of the form y n = y n ( p , a )= ~ k =0 X n kp a k where p is an arbitrary prime. Let 𝜌 p ( m ) denote the exponent of the highest power of a prime p which divides m . Recently, the author obtained lower bounds for 𝜌 p ( y n ) based on recurrence relations of order p and p m 1. The cases with tight bounds have also been characterized. In this paper, we prove that 𝜌 p ( y np ( p , a ))= n for 𝜌 p ( a +1)=1, p S 3. We obtain alternative difference equations of order p 2 for y n and order p for the p -sected sequence y np by a generating function based method. We also extend general divisibility results relying on the arithmetic properties of the coefficients and initial values.     

一种用于评价集体成绩的序列加权均值

本文对评价中通常惯用的平均化方法进行了反思,并结合现实问题,提出了一种利用混合排序进行赋权的序列加权均值评价法。这种方法的基础来自于算术平均与加权平均的思想,其观念是鼓励先进,对优秀个体所做贡献的价值进行适当的提升。本文还通过一个简单算例,与传统方法进行了对比分析,结论是可信的。     

Gorenstein投射、内射和平坦复形

证明了在任意结合环R上,复形C是Gorenstein投射复形当且仅当每个层次的模C~m是Gorenstein投射模,由此给出了复形Gorenstein投射维数的性质刻画.并证明了对于正合复形C,若对于任意投射模Q,函子Hom(-,Q)作用复形C后仍然得到正合复形,则C是Gorenstein投射复形当且仅当对于所有的m∈Z,有Ker(δ_C~m)是Gorenstein投射模.类似地,本文也讨论了关于Gorenstein内射和Gorenstein平坦复形的相应结果.     

关于奇算术图

本文讨论了$n$个$m$长圈有一个公共结点图$C^n_m$, $n$个$m$长圈与$t$长路有一个公共结点图$C^n_m\cdot P_t$, $n$个$m$阶完全图有一个公共结点图$K^n_m$和星形图的同胚图的奇算术性问题.给出了完全图,完全二部图和圈是奇算术的充要条件.     

Evolutionary learning of fuzzy grey cognitive maps for the forecasting of multivariate,interval-valued time series

Time series are built as a result of real-valued observations ordered in time; however, in some cases, the values of the observed variables change significantly, and those changes do not produce useful information. Therefore, within defined periods of time, only those bounds in which the variables change are considered. The temporal sequence of vectors with the interval-valued elements is called a ‘multivariate interval-valued time series.’ In this paper, the problem of forecasting such data is addressed. It is proposed to use fuzzy grey cognitive maps (FGCMs) as a nonlinear predictive model. Using interval arithmetic, an evolutionary algorithm for learning FGCMs is developed, and it is shown how the new algorithm can be applied to learn FGCMs on the basis of historical time series data. Experiments with real meteorological data provided evidence that, for properly-adjusted learning and prediction horizons, the proposed approach can be used effectively to the forecasting of multivariate, interval-valued time series. The domain-specific interpretability of the FGCM-based model that was obtained also is confirmed.     

Homogenization and corrector for the wave equation with discontinuous coefficients in time

In this paper we analyze the homogenization of the wave equation with bounded variation coefficients in time, generalizing the classical result, which assumes Lipschitz-continuity. We start showing a general existence and uniqueness result for a general sort of hyperbolic equations. Then, we obtain our homogenization result comparing the solution of a sequence of wave equations to the solution of a sequence of elliptic ones. We conclude the paper making an analysis of the corrector. Firstly, we obtain a corrector result assuming that the derivative of the coefficients in the time variable is equicontinuous. This result was known for non-time dependent coefficients. After, we show, with a counterexample, that the regularity hypothesis for the corrector theorem is optimal in the sense that it does not hold if the time derivative of the coefficients is just bounded.     

一般半相依回归系统的协方差改进估计

本文讨论了由两个等阶的回归方程组成的半相依系统,运用协方差改进法获得了参数的一个迭代估计序列,并证明了它的协方差阵已知时,处处收敛到最佳线性无偏估计,同时其协方差阵在矩阵偏序意义下单调性,并且给出了当迭代次数亦趋于无穷时,保证其具有相合性的一个条件。     

On Finite Pseudorandom Binary Sequences, VI,(On Sequences)

 Let K be a positive integer and α be a real number, and for let if the fractional part of is , and if it is . The pseudorandom properties of the sequence are studied. As measures of pseudorandomness, the regularity of the distribution relative to arithmetic progressions and the correlation are used. In a previous paper the authors studied the special cases and , while here the case is considered.     

关于凝聚局部环的正则性

本文证明了极大理想m是有限生成的交换凝聚局部环（R,M）是正则的充分必要条件是m可以由一个正则R-序列生成,推广了文献[1]中相应的结论并给出了一个由正则凝聚局部环构造大量的非正则凝聚局部环的方法．